# Full text of "Treatise On Analysis Vol-Ii"

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```132       XIII    INTEGRATION

5.    Let A be Lebesgue measure on R, and let E be the topological vector space obtained
by endowing the space ^(X) with the topology of simple convergence (12.15). Show
that the linear form/V~»J/dA on E is not continuous.

6.    Let (An) be a sequence of /x-integrable subsets of X, such that £ ft(Aw) < + 00. For

n

each integer k, let Gfc be the set of all x e X such that x e An for at least k values of n.
Show that Gfc is integrable and that k • /x(G*)

7.    Let X be a compact space, \L a positive measure on X, and (An) a sequence of integrable
subsets of X such that inf />t(An) = m > 0. Show that the set B of points of X which

n

belong to infinitely many of the sets An is integrable, and that /x(B) ^ m.

8.    Let A be Lebesgue measure on the interval I = [0, 1] and let Aw be a sequence of
integrable sets in I such that inf A(A«) == m > 0. For each integer k, let \$)k denote the

set of 2k intervals of the form [j - 2~fc, (/ + I) • 2"k] in I (0 ^/ < 2*), and <g* the set of
all the unions of sets of 25*. Show that there exists a decreasing sequence (10*20 of
subsets of I such that I0 = I, I* e @*for all k, and a subsequence (Anfc) of the sequence
(An) with the following properties: (i) A(Anr n (I - I*)) ^ JwA(I ~Ifc) for all r ^ k);
(ii) A(A«r n J) ^ £wA(J) for all r ^ k and all J e 5D* contained in I* . (Proceed by in-
duction on k, using the diagonal trick.) Deduce that there exists a subsequence (Brt)
of (An) such that, for all k and all J e 3)fc contained in I* , the intersection of J with
the sequence (B«) is not empty. (Use Problem 7 and the diagonal trick.) Deduce
that p| Bn contains a non-empty compact set with no isolated points (and therefore

n

nondenumerable (Section 4.2, Problem 3(c))).

9.    Let /be a real- valued function ^0 on X. Show that the mapping //, i— » ju,*(/) of M +(X)
into R is continuous with respect to the vague topology (13.4) if and only iff is con-
tinuous and compactly supported. The mapping /AI— >/x*(/) is lower semicontinuous
with respect to the vague topology if and only if /is lower semicontinuous.

10.    Let (/z,n) be an increasing sequence of positive measures on X. Suppose that the se-
quence is bounded above in M+(X), and let ^ be its least upper bound (13.4.4).
Show that, for every function/^ 0 on X, we have fi*(/) = lim /*?(/).

n-»-oo

11.    (a)   Let (//.„) be a decreasing sequence of positive measures on X, and let /z be the
greatest lower bound of the sequence in M+(X) (13.4.4). If /is a function ^>0 on X
such that ju*(/) < + oo for all sufficiently large n, show that ^*(/) = lim /^(/).

n-*oo

(If g ^ 0 is lower semicontinuous and n*(g) < + oo, observe that there exists a se-
quence (/zm) of continuous functions ^0 with compact supports, such that]£ hm <*g

and p*(ff) «£ p*(hm) for alt n.)

m

(b) On the discrete space N, let //,„ be the measure such that /nn({w}) = 0 for m < n
and 1 for m ^ n. Show that the greatest lower bound of the decreasing sequence (//.„)
in M+(N) is 0, but that /z*(N) = + oo for all n. Fx. If K is the convex and vaguely compact set of positive measures of
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